linear algebra proof help!

Let S = {V1,ůvn} be a linearly independent set in a vector space V. Suppose w is in V, but w is not in span(S). Prove that the set T = {V1,ůVn,w} is a linearly independent set.

since it says w is in v, does it mean that w is a subspace of v? yet w is not in span(S). I am kinda confused with what it means?

I am thinking that suppose w is not = 0, and because w is in V.
a1v1+a2v2+...+anvn+a(n+1)w=0 which leads to a(n+1)=-a1v1-a2v2-...anvn
and then because S={V1,ůvn} is a linearly independent set. a1v1, a2v2... are all zeros, so {V1,ůVn, W} would be linearly indep too. then w can be write in a linear combi. thus T is a linearly indep set.

Say that , there are two cases. Either or . If then we end up with and so , which shows that is linearly independent. Now say that . In this case divide by to get: where . Thus, which is a contradiction.